LP-relaxations for tree augmentation

In the Tree Augmentation problem the goal is to augment a tree T by a minimum size edge set F from a given edge set E such that T∪F is 2-edge-connected. The best approximation ratio known for the problem is 1.5. In the more general Weighted Tree Augmentation problem, F should be of minimum weight. Weighted Tree Augmentation admits several 2-approximation algorithms w.r.t. the standard cut-LP relaxation. Improving this natural ratio is a major open problem, and resolving it may have implications on other network design problems. It seems that achieving this goal requires finding an LP-relaxation with integrality gap better than 2, which is an old open problem even for Tree Augmentation. In this paper we introduce two different LP-relaxations, and for each of them give a simple combinatorial algorithm that computes a feasible solution for Tree Augmentation of size at most 1.75 times the optimal LP value.